NUMERICAL CONVERSION OF TRANSIENT TO HARMONIC RESPONSE FUNCTIONS FOR LINEAR VISCOELASTIC MATERIALS

Viscoelastic material behavior is often characterized using one of the three measurements: creep, stress-relaxation or dynamic sinusoidal te sts. A two-stage numerical method was developed to allow representatio n of data from creep and stress-relaxation tests on the Fourier axis i n the Laplace domain. The method assumes linear behavior and is theore tically applicable to any transient test which attains an equilibrium state. The first stage numerically resolves the Laplace integral to co nvert temporal stress and strain data, from creep or stress-relaxation , to the stiffness function, G(s), evaluated on the positive real axis in the Laplace domain. This numerical integration alone allows the di rect comparison of data From transient experiments which attain a fina l equilibrium state, such as creep and stress relaxation, and allows s uch data to be fitted to models expressed in the Laplace domain. The s econd stage of this numerical procedure maps the stiffness function, G (s), from the positive real axis to the positive imaginary axis to rev eal the harmonic response function, or dynamic stiffness, G(j omega). The mapping for each angular frequency, s, is accomplished by fitting a polynomial to a subset of G(s) centered around a particular value of s, substituting js for s and thereby evaluating G(j omega). This two- stage transformation circumvents previous numerical difficulties assoc iated with obtaining Fourier transforms of the stress and strain time domain signals. The accuracy of these transforms is verified using mod el functions from poroelasticity, corresponding to uniaxial confined c ompression of an isotropic material and uniaxial unconfined compressio n of a transversely isotropic material. The addition of noise to the m odel data does not significantly deteriorate the transformed results a nd data points need nor be equally spaced in time. To exemplify its po tential utility, this two-stage transform is applied to experimental s tress relaxation data to obtain the dynamic stiffness which is then co mpared to direct measurements of dynamic stiffness using steady-state sinusoidal tests of the same cartilage disk in confined compression. I n addition to allowing calculation of the dynamic stiffness from trans ient tests and the direct comparison of experimental data from differe nt tests, these numerical methods should aid in the experimental analy sis of linear and nonlinear material behavior, and increase the speed of curve-fitting routines by fitting creep or stress relaxation data t o models expressed in the Laplace domain. Copyright (C) 1996 Elsevier Science Ltd.